1. Field of the Invention
The invention relates to a method and apparatus for generating consistent image registration. More specifically, the invention relates to a method and apparatus for generating inversely related forward and reverse image transformations in any image registration technique with ambiguous correspondence.
2. Description of Prior Art
A reasonable but perhaps not always desirable assumption is that the mapping of one anatomical image (source) to another (target) is diffeomorphic, i.e., continuous, one-to-one, onto, and differentiable. A diffeomorphic mapping has a unique inverse that maps the target image back onto the source image. Thus, it is reasonable goal to estimate a transformation from image A to B that should equal the inverse of the transformation estimated from B to A assuming a diffeomorphic mapping exists between the images. However, this consistency between the forward and reverse transformations is not guaranteed with many image registration techniques.
Depending on the application, the diffeomorphic assumption may or may not be valid. This assumption is valid for registering images collected from the same individual imaged by two different modalities such as MRI and CT, but it is not necessarily valid when registering images before and after surgery. Likewise, a diffeomorphic mapping assumption may be valid for registering MRI data from two different normal individuals if the goal is to match the deep nuclei of the brain, but it may not be valid for the same data sets if the goal is to match the sulcal patterns.
Alternatively, diffeomorphic transformations may be used to identify areas where two image volumes differ topologically by analyzing the properties of the resulting transformation. For example, consider the problem of matching an MRI image with a tumor to one without a tumor. A possibly valid diffeomorphic transformation would be one that registers all of the corresponding brain structures by shrinking the tumor to a small point. Such a transformation would have an unusually small Jacobian, which could be used to detect or identify the location of the tumor. Conversely, consider the inverse problem of matching the image without the tumor to the one with the tumor. A valid registration in this case may be to register all of the corresponding brain structures by allowing the transformation to xe2x80x9ctearxe2x80x9d (i.e., not be diffeomorphic) at the site of the tumor. Just as valid could be a diffeomorphic transformation that registers all of the corresponding brain structures by allowing the transformation to stretch at the site of the tumor.
As in the previous examples, the assumption can be made that a valid transformation is diffeomorphic everywhere except possibly in regions where the source and target images differ topologically, e.g., in the neighborhood of the tumor. These ideas can be extended to certain non-diffeomorphic mapping problems by including boundary conditions to model, isolate or remove regions that differ topologically.
Transformations that are diffeomorphic maintain topology guaranteeing that connected subregions remain connected, neighborhood relationships between structures are preserved, and surfaces are mapped to surfaces. Preserving topology is important for synthesizing individualized electronic atlases; the knowledge base of the atlas maybe transferred to the target anatomy through the topology preserving transformation providing automatic labeling and segmentation. If total volume of a nucleus, ventricle, or cortical sub region are an important statistic it can be generated automatically. Topology preserving transformations that map the template to the target also can be used to study the physical properties of the target anatomy such as mean shape and variation. Likewise, preserving topology allows data from multiple individuals to be mapped to a standard atlas coordinate space. Registration to an atlas removes individual anatomical variation and allows information from many experiments to be combined and associated with a single conical anatomy.
The forward transformation h 130 from image volumes T 110 to S 120 and the reverse transformation g 140 from S 120 to T 110 are depicted in FIG. 1. Ideally, the transformations h 130 and g 140 should be uniquely determined and should be inverses of one another. Estimating h 130 and g 140 independently very rarely results in a consistent set of transformations due to a large number of local minima. To overcome this deficiency in current registration systems, the present invention jointly estimate h 130 and g 140 while constraining these transforms to be inverses of one another. The invertibility constraint will reduce the number of local minima because the problem is being solved from two different directions. Although uniqueness is very difficult to achieve in medical image registration, the joint estimation should lead to more consistent and biologically meaningful results.
According to the teachings of the present invention, more meaningful results may be achieved over prior art registration methods and systems through registering images based upon joint estimation forward and reverse transformations. In this joint estimation, two image data sets are received, a source and a target. A current set of forward and reverse displacement fields are initialized. From the current set of forward and reverse displacement fields, the consistent forward and reverse transformations are jointly estimated. These transformations are outputted.
In one embodiment, the joint estimation of the forward and reverse transformation is based upon utilization of a symmetric cost function. In further embodiments, the symmetric cost function is combined with either an inverse consistency constraint function or a diffeomorphic constraint function to perform the joint estimation. In yet another embodiment, the symmetric cost function is combined with both an inverse consistency constraint function and a diffeormorphic constraint function to serve as a basis for the joint estimation of the forward and reverse transformations. In yet another embodiment, the diffeomorphic-constraint can be replaced by or combined with another regularization constraint that maintains desirable properties of the template (source) and target images when deformed. In yet another embodiment, all of the previously stated constraints can be combined with constraints that enforce correspondence between corresponding landmarks, contours, and surfaces in the template and target images.